Newspace parameters
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 11.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(8.45177498616\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.202533.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 37x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 37x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} + 6\nu - 52 \)
|
| \(\beta_{2}\) | \(=\) |
\( -4\nu^{2} + 12\nu + 96 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 8 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 2\beta _1 + 200 ) / 8 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−75.6271 | −281.520 | 3671.46 | 5111.20 | 21290.6 | 21831.1 | −122777. | −97893.2 | −386545. | |||||||||||||||||||||||||||
| 1.2 | 23.3081 | 296.237 | −1504.73 | −13329.8 | 6904.73 | 32948.8 | −82807.5 | −89390.6 | −310692. | ||||||||||||||||||||||||||||
| 1.3 | 52.3190 | −407.717 | 689.274 | 913.580 | −21331.3 | −59861.9 | −71087.1 | −10914.2 | 47797.6 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 11.12.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 99.12.a.a | 3 | ||
| 4.b | odd | 2 | 1 | 176.12.a.e | 3 | ||
| 5.b | even | 2 | 1 | 275.12.a.a | 3 | ||
| 11.b | odd | 2 | 1 | 121.12.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.12.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 99.12.a.a | 3 | 3.b | odd | 2 | 1 | ||
| 121.12.a.c | 3 | 11.b | odd | 2 | 1 | ||
| 176.12.a.e | 3 | 4.b | odd | 2 | 1 | ||
| 275.12.a.a | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4500T_{2} + 92224 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(11))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 4500T + 92224 \)
$3$
\( T^{3} + 393 T^{2} + \cdots - 34002261 \)
$5$
\( T^{3} + 7305 T^{2} + \cdots + 62243294875 \)
$7$
\( T^{3} + 5082 T^{2} + \cdots + 43059126844184 \)
$11$
\( (T - 161051)^{3} \)
$13$
\( T^{3} + 2434212 T^{2} + \cdots + 33\!\cdots\!56 \)
$17$
\( T^{3} - 12112122 T^{2} + \cdots + 21\!\cdots\!48 \)
$19$
\( T^{3} + 8590560 T^{2} + \cdots - 16\!\cdots\!00 \)
$23$
\( T^{3} + 3136413 T^{2} + \cdots + 14\!\cdots\!59 \)
$29$
\( T^{3} + 376441824 T^{2} + \cdots + 18\!\cdots\!40 \)
$31$
\( T^{3} + 313174893 T^{2} + \cdots + 69\!\cdots\!75 \)
$37$
\( T^{3} + 454281387 T^{2} + \cdots - 98\!\cdots\!23 \)
$41$
\( T^{3} + 37614456 T^{2} + \cdots + 60\!\cdots\!08 \)
$43$
\( T^{3} - 162163386 T^{2} + \cdots + 19\!\cdots\!72 \)
$47$
\( T^{3} + 3182498184 T^{2} + \cdots - 45\!\cdots\!28 \)
$53$
\( T^{3} + 3000753402 T^{2} + \cdots - 30\!\cdots\!72 \)
$59$
\( T^{3} + 1843219707 T^{2} + \cdots - 26\!\cdots\!15 \)
$61$
\( T^{3} + 28094112684 T^{2} + \cdots + 79\!\cdots\!68 \)
$67$
\( T^{3} - 10315312497 T^{2} + \cdots + 86\!\cdots\!09 \)
$71$
\( T^{3} - 3703071657 T^{2} + \cdots - 11\!\cdots\!39 \)
$73$
\( T^{3} - 14017034988 T^{2} + \cdots + 49\!\cdots\!76 \)
$79$
\( T^{3} + 8104583058 T^{2} + \cdots + 20\!\cdots\!20 \)
$83$
\( T^{3} - 26009027946 T^{2} + \cdots + 54\!\cdots\!32 \)
$89$
\( T^{3} + 17344395051 T^{2} + \cdots - 22\!\cdots\!15 \)
$97$
\( T^{3} + 7984545237 T^{2} + \cdots - 28\!\cdots\!93 \)
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